16 
applied to an earlier version of the data-set used in this work. For a detailed description we 
refer to Gouretski and Jancke (2001), and below is given only an overview of the method. 
The procedure starts with the estimation of inter-cruise property offsets within cross-over 
areas. The size of the crossover area and the potential temperature and depth ranges were 
specified based on the estimate of parameter variability on potential temperature surfaces. 
The size of the crossover area was limited by 300 km, and only samples below 800 meters 
and colder than 3°C were taken for the inter-comparison. For each pair of intersecting 
cruises/sections the inter-cruise offset is calculated as the average of individual profiles. 
Errors in temperature are neglected. A number of geographical areas with extremely high 
variability within the deep part of the water column were excluded (e.g. Irminger and Labrador 
Sea, Antarctic continental slope). 
Gouretski and Jancke (2001) suggested a decomposition of observed inter-cruise offsets D into 
systematic and non-systematic components. For each pair of cruises (i,j) the observed offset is: 
D/j = Ay + rijj, (1) 
where = <5, - <5, is the true offset, e.g. the part of the inter-cruise offset due to systematic 
errors <5 in the data, whereas /% represents the non-systematic part of the offset, caused by 
random errors and by the combined effect of the time-space variability within the cross-over 
area. 
3.5. Calculation of biases for reference cruises 
Typically, for a set of N cruises the number of the offset estimates M » N (M is the number of 
crossover areas). The set of equations (1) may be written in the general standard form : 
ES + n = D, (2) 
where in the present case, solution, noise and offset vectors are: 
S = (S h 82, 8n) 
n = (n u n 2 ,n N ) 
D = (D-t, D 2 ,.,Dm). 
For a pair of cruises (p,q) elements of the k-th row of the matrix E are: 
E w 
1 for I = p, 
-1 for I = q , 
0 for I * p, I * q 
(3) 
(4) 
(5) 
(6) 
Thus, the M equations (2) are used to estimate N values 8 and M values n i; or M+N altogether. 
The solution of the system is obtained in a root-mean square sense (Wunsch, 1996): 
8=(E T E)' 1 E T D; (7) 
For the reference data set the system of equations (2) was solved to get the reference cruise 
biases 5. Since no true reference data is available, we subtracted the average of all WOCE 
biases from each individual bias after the root-mean-square solution of (11) was obtained. The 
total number of equations (cruise pairs) was of 2094, 665, 404, 331 and 331 for salinity, oxygen, 
silicate, nitrate, and phosphate respectively, with systematic biases 5 determined for 384, 213, 
145,133 and 136 cruises.